1. Field of the Invention
The present invention relates to a method of plasma particle simulation, a storage medium, a plasma particle simulator and a plasma processing apparatus, and more particularly, to a method of plasma particle simulation for plasma with a sheath.
2. Description of the Related Art
A plasma processing apparatus for subjecting a plasma processing to a wafer or the like using plasma includes a housing chamber for housing the wafer. Within the housing chamber, there is generated plasma. Since a plasma distribution has a significant effect on the uniformity of processing subject to the wafer, many attempts have been heretofore made in order to know the distribution of plasma within the housing chamber.
As a method for directly observing the plasma distribution, there is known a method for observing a state of light emission within the housing chamber. Since being based on optical methodology, the aforementioned method has many factors of disturbance and, hence, a poor accuracy of observation. Accordingly, a multitude of simulations has been run in recent years, in order to predict a distribution of plasma particles and a temperature distribution by calculating the behavior of plasma particles (radicals, positive ions, electrons and the like) using a computer. In these simulations, there is often used a calculation technique known as the Monte Carlo method.
The Monte Carlo method is a calculation technique used to determine numerical solutions by stochastically solving equations. In particular, there is suitably used in plasma particle simulation, the Particle-in-Cell, Monte-Carlo-Collision (PIC-MCC) method in which the behavior of each plasma particle is mathematized, a space within the housing chamber as an objective space to be calculated is divided into a plurality of cells, and the behavior calculation of each plasma particle in each cell is statistically performed on the basis of mathematical expressions.
On the other hand, the number of plasma particles contained in the housing chamber is enormous. Consequently, performing a behavior calculation for each of plasma particles requires an enormous computational load, thus taking a vast amount of time for simulation. Hence, a concept called a superparticle which represents a large number of plasma particles (for example, 107 particles) is applied in the simulation of plasma particle behavior using the Monte Carlo method. Here, the number of plasma particles represented by one superparticle is referred to as a “weighting factor.” In this case, a behavior calculation is made for each superparticle rather than for each plasma particle. Thus, the calculation load (memory and time) and calculation cost are prevented from increasing to vast amounts (see, for example, Japanese Laid-Open Patent Publication (Kokai) No. 8-235156).
Incidentally, the housing chamber of the processing apparatus is generally specified as taking on a cylindrical shape in the simulation of plasma particle behavior. Consequently, if the space within the housing chamber is divided axisymmetrically with respect to the central axis thereof into a plurality of cells, the volume of a cell near the central axis becomes small (see FIG. 10). Assuming at this point that the density of plasma particles in the space is constant, then the number of plasma particles contained in a cell near the central axis decreases. As a result, the number of superparticles subject to behavior calculation also decreases. On the other hand, the Monte Carlo method has the problem that a statistical error in calculations becomes larger as the number of superparticles decreases.
What is required in order to control the statistical error is to increase the number of superparticles contained in a cell. In response to this requirement, there has been proposed a method for maintaining the number of superparticles contained in a cell near the central axis even if the number of plasma particles decreases, by making the weighting factor smaller as the cell is located closer to the central axis (see, for example, Hideto TAKEIDA and Kenichi NANBU, “Weighting factor for Particle Modeling of Axisymmetrical Low Temperature Plasma”, Journal of the Physical Society of Japan, Vol. 73, No. 3, The Physical Society of Japan, Mar. 2004, p. 756-757). In addition, a method for maintaining the number of superparticles contained in cells near the central axis by applying an extremely small weighting factor to every cell, has also been implemented by the present inventor et al. (see, for example, Kazuki DENPOH and Kenichi NANBU, “Self-consistent particle simulation of radio-frequency CF4 discharge with implementation of all ion-neutral reactive collisions”, Journal of vacuum Science & Technology A, American Vacuum Society, May/June 1998, p. 1201-1206).
However, sheaths are produced in the vicinity of solid wall surfaces (e.g., sidewall and ceiling wall) of the housing chamber when plasma is generated therein. A sheath is a region which is produced due to a difference between electrons and positive ions in the speed of arrival at a solid wall surface and in which the density of plasma particles, particularly electrons, is low. Accordingly, if a real plasma particle distribution is reflected in the simulation of plasma particles, the number of plasma particles contained in a cell near a solid wall surface becomes small in a case where the space within the housing chamber is divided into a plurality of cells.
Furthermore, since a potential difference and an electric field gradient are large in a sheath, a portion where the sheath is produced (portion near the solid wall surface) generally needs to be divided into a large number of cells, in order to improve calculation accuracy. This means that the number of plasma particles contained in a cell near the solid wall surface also becomes small.
As described above, the number of superparticles subject to behavior calculation also decreases in the cell near the solid wall surface and, therefore, a statistical error in behavior calculation becomes large. As a result, it is no longer possible to ensure consistency with the results of behavior calculation of superparticles in the portion (bulk) of plasma other than the sheath, possibly causing a solution to diverge in plasma particle simulation.